Group homomorphism/Matrix 3 4 1 2/Bijective for Q, not for Z/Exercise

We consider the matrix

${\begin{pmatrix}3&4\\1&2\end{pmatrix}}.$

Show that this matrix defines a group homomorphism from ${}\mathbb {Q} ^{2}$ to ${}\mathbb {Q} ^{2}$, and from ${}\mathbb {Z} ^{2}$ to ${}\mathbb {Z} ^{2}$ as well. Study this group homomorphism with respect to injectivity and surjectivity.

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